KnowraEven numberEven numberAn even number is an integer divisible by 2, equivalently an integer of the form 2k for some integer k.BriefConnectInteger: A whole number that may be positive, negative, or zero. Evenness is defined for integers, not for arbitrary real numbers.Modular arithmetic: Arithmetic in which numbers are grouped by their remainders modulo a fixed integer. Calculating modulo 2 reduces parity questions to two residue classes.Parity argument: A proof technique that derives a contradiction or conclusion from whether integers are even or odd. Parity can rule out integer solutions without calculating their exact values.Prime number: A positive integer greater than 1 whose only positive divisors are 1 and itself. Every even integer greater than 2 is composite, leaving 2 as the sole even prime.Divisibility: A relation in which one integer is a multiple of another integer. An integer is even precisely when 2 divides it.Odd number: An integer that leaves remainder 1 when divided by 2. Odd integers are the other residue class modulo 2.Pigeonhole principle: A counting principle stating that placing more objects than containers forces some container to hold multiple objects. Parity can sharpen counting arguments by separating objects into two classes.Composite number: A positive integer greater than 1 that has a positive divisor other than 1 and itself. Evenness alone does not imply compositeness because 2 is prime.Division algorithm: A theorem expressing an integer as a divisor times a quotient plus a bounded remainder. Division by 2 leaves remainder 0 for even integers and 1 for odd integers.Parity of a sum: The rule determining whether a sum is even or odd from the parities of its addends. Two even integers always have an even sum, while adding an odd integer changes parity.Show all 24Linked from 7 pagesIf and only ifBroader topic: Its standard definitions are equivalent in both directions.Goldbach's conjectureRelated: Evenness is the precise condition on the integers covered by the conjecture.Euclid–Euler theoremNarrower topic: The theorem applies only to perfect numbers divisible by two.Set-builder notationRelated: The set of even integers is defined by the condition that an integer equals twice another integer.Goldbach–Euler theoremRelated: The conjecture concerns every even integer greater than 2.Lemoine's conjectureRelated: Twice a prime is always even, so subtracting it leaves an odd prime candidate.Von Staudt–Clausen theoremRelated: The theorem applies to Bernoulli numbers with positive even index 2n.